Aircraft: KODIAK 100
Phase of flight: Final Approach
Anomaly: ATC Issue All Types; Conflict NMAC; Deviation / Discrepancy - Procedural Published Material / Policy

The pilot; 'Pilot 1' of a single-engine turboprop aircraft was on approach to a Class C airport from the north. Pilot 1 was operating under Part 91 VFR rules in day VMC conditions. Per ATIS; visibility was 10 miles and ceiling clear below 12;000 ft. Pilot 1 was handed off from Approach Control to Tower and was instructed by Tower to 'continue' to Runway XXR. A landing clearance had not yet been issued. While descending through approximately 6;000 ft. MSL (3;000 ft. AGL) Pilot 1 experienced a rapid onset of severe pain in his sinus. When reaching approximately 5;000 ft. MSL; Pilot 1 leveled off and initiated a slow climb back to 6;000 ft. MSL. Tower frequency was busy as Tower appeared to be giving lengthy instruction to a flight of aircraft while also managing other aircraft within the airspace. Pilot 1 was unable to 'break in' and talk with Tower. Without being able to coordinate with Tower; Pilot 1 elected to discontinue approach and turn away from the airport (after crossing through the final approach corridor) and maintain approximately 6;000 ft. MSL on a heading which was 20-30 degrees divergent from inbound runway traffic in an effort to maintain separation from other aircraft; and also manage the physiological issue (equalize pressure in the sinus). After approximately 1-2 minutes of flying away from the airfield; Tower queried Pilot 1 as to his intentions. Pilot 1 stated that he did not require assistance at the time but did need and altitude of 6;000 ft. MSL and a heading to clear airspace. Pilot 1 was then given a heading of 190 degrees (this would have been a safe heading). Within seconds of receiving the 190 heading; Pilot 1 was told by Tower 'Traffic alert. Turn left heading 360 immediately!' As Pilot 1 was executing this turn; he and his passenger visually spotted a Part 121 B737. The assigned 360-degree heading had position Pilot 1 directly in front of and slightly below the 737's flight path. Pilot 1 was forced to take evasive action and aggressively maneuver away from; and below the 737. The crew of the 737 received and reported (to Tower) a resolution advisory. The crew of the 737 stated (over Tower frequency) that they had visual contact with Pilot 1's aircraft as it was maneuvering away from them.After the traffic conflict was resolved; Pilot 1 again requested 6;000 ft. MSL and a vector to clear airspace which was accommodated by Tower. Pilot 1 ultimately made a safe landing. Pilot 1 did not [request priority handling] during this incident. The near mid-air collision took place approximately 7 miles east of the airfield runway approach end. No aircraft damage or injury resulted from this incident.Contributing factors [include] lengthy communication between Tower and aircraft on the ground while simultaneously managing several airborne aircraft. Per the Tower Supervisor; Tower's radar has a 6-second sweep. Thus; in this scenario radar provided delayed information to the controller resulting in instruction of a turn of Pilot 1 in an unsafe direction. [Also contributing was] single pilot operation in which pilot was managing [their] own physiological emergency while simultaneously managing safety of the flight.At 7 miles; the two incident aircraft were likely beyond visual range of the Tower controller(s). This airfield can be a busy Class C with a mix of General Aviation; Military; and Air Carrier operations. [I] recommend upgrading Tower radar to one with a faster refresh rate; assigning an alternate tower frequency where lengthy discussion (preferably with an alternate controller) can be had; [and] that flight crew members carry fast acting decongestant. Kodiak 100 Pilot reported a NMAC event while attempting to maneuver away from approach course via an assigned heading.